Not having internet has been annoying since so much has been discussed in this book, and I wanted to get my ideas out before I forgot them. I tried jotting notes in the margins, but this post won’t do it justice. This will conclude the material of the book. There will still be a few more discussing implications that I’ve thought of while synthesizing the material. Here goes:

I’m still on the fence about the novelty of the idea of the BMI. This is the case even after many examples. It just seems that mathematicians already know they are doing this. Luckily, I am completely reverse of the reviews of this book. I think the first half was so-so, but it starts to kick out some great ideas in the latter half.

Chapter 9: Real Numbers and Limits. Some interesting commentary on why epsilon-delta definitions are difficult for students. Limits are conceptualized using the motion along a line metaphor for numbers, yet the definition is very static. It also accounts for ridiculous cases of epsilon (who cares about large epsilon? it is only the small epsilon that matter in limits). This is mathematically a necessity, but conceptually confusing. I agree. On to sums. Hmm…still not all that interesting. Just examples of the use of the BMI.

Chapter 10: Transfinite numbers. They talk about Cantor’s diagonal argument and some of the assumptions. The proof is usually taken to be formal, but in actuality it cannot be written down formally because you can’t express infinity as an actual entity. This means that the fact that there are more reals than rationals is inherently metaphorical. It is also discussed that Cantor’s one-to-one and onto definition for equivalent infinities is metaphorical and not absolute. It is one way to count infinities and see if they are the same. It is not the only way. We often lose sight of this, or not even realize it.

Note: Getting the picture yet? They are building this idea of metaphor to a pretty interesting climax.

Chapter 11: Infinitesimals. This was probably my favorite chapter because I had never seen a construction of the hyperreal numbers. They first builds what they call the granular numbers. This is essentially just the first “layer” of the hyperreals. You get the interesting result that I didn’t know about that there are number systems with no possible system of numerals (because it would need an infinite alphabet to express). This also brings up the concept that there are mathematical objects that are inherently metaphorical (since they can’t be expressed otherwise). I think that they think the most important part of this chapter is that “ignoring certain differences is absolutely vital to mathematics.” This refutes the idea of mathematics being perfect, exact, absolute, … , i.e. Platonistic. Yea! Finally, the big one comes out. This argument is much longer, but basically boils down to “calculus is defined by ignoring infinitely small differences.”

Chapter 12: Points and the Continuum. What to say about this… Basically it goes through the struggle of how to define a point. Do points on the real line touch? If they do, then by definition of having no length they are the same point. So they can’t touch. But the real line is continuous, i.e. there are no gaps. Thus points much touch. A paradox? Actually they break this down as a problem of blending two metaphors for talking about . This shows that when we talk about things as absolute truth, we may actually be referring to a metaphor which doesn’t exactly work in every situation. We must be careful what metaphor we are using and how it affects what we are talking about. Also the problem of attempting to discretize (write down mathematics in a precise and logical manner from axioms) the continuous is discussed. From a conceptual point of view this is impossible. In fact, it really hasn’t been insanely successful.

Chapter 13: Continuity for Numbers: The Triumph of Dedekind’s Metaphors. This talks about Dedekind cuts. Blah.

Chapter 14: Calculus Without Space or Motion: Weierstrass’s Metaphorical Masterpiece. This talks about how the geometric interpretation/metaphor for calculus had major limitations. There were functions that had nothing to do with motion. It talks about how Weierstrass extended calculus to work in these situations. Here again is the “choice of metaphor argument.”

Le Trou Normand. Here they give us the kicker. I’m going to do this in a more concise way. Construct the sequence of functions semicircles of perimeter (where the first one starts at and the last one ends at . Now each has arclength , but the sequence of functions converges to [0,1]. There is an apparent contradiction since the arclengths of the sequence converges to and thus $[0,1]$ has “length” .

The problem is the same as before. Our choice of metaphor is incorrect. We can’t say that the limit of the length of a pointwise convergent sequence of functions is the length of the limit under our current metaphor. But we can define a new metaphor in which this works. This is a common metaphor to use in functional analysis. Construct a function space in which our distance is . You can work this stuff out for yourself to see how it works.

Moral: Our choice of metaphor matters! Down with Platonism! We can’t treat functions as literally being curves in the plane or the motion of a particle. While these are useful metaphors at times, they should not be taken as literal objective representations that give us all the information and no excess incorrect information (careful on all the negatives I stuck in there).

Tomorrow: Philosophical implications.

Re Chapt 12,

But the real line is continuous, i.e. there are no gaps. Thus points much touch.Or, there are no gaps so the line is not put together out of points like a string of beads out of beads (since then they would touch).Some people say that because the points have zero size (unlike beads) hence they can form a continuum without having to touch, but while it is hard to put one’s finger on the contradiction (maybe we just expect that no gaps implies they are touching insofar as we wrongly think of them as like beads) we do seem to need of a way of thinking of these inconceivably small things. We need something like a metaphor, or at least a story. The following is my story:

So I say (apologies for being a Platonist) that the line is primary. (The concept of spatial extension is primitive, anyway.) Then we can imagine a point on the line as being where two lines cross. It is common to both lines; it is a part of each line. (Lines are too thin to visualise properly, but we can visualise a point as being the location of the corner of a square.) Then we can imagine that such points would be everywhere in the line, since nowhere in the line would be too small to have points there.

So the line is

fullof points, if notmadeof points. They don’t touch, but because they are not like beads, we don’t need them to. And there are no gaps because points are so small. Anyway, that’s what I think about points; what do you think? Do I avoid paradox? (I avoid some problems by thinking of the line as quasi-spatial, and the transfinite collections as quasi-temporal, as indefinitely extensible, but that’s another story.)This is almost precisely what the authors go into (I didn’t write it since I was trying to cram so much in). You can use the metaphor that the line is by nature continuous (not the number line, but a gapless line). Then you fill in the line with real numbers. We are mixing metaphors when we say that the real numbers are the line and not the real numbers fill in a naturally continuous line. So I think the authors would agree with you.

This pushes us a little further, though. The set of rational numbers placed on a naturally continuous line is continuous as well. (Within the set of rational numbers there is no such thing as an irrational and thus no gaps can exist). This is the same thing as saying that within the set of reals no hyperreals exist. If they did, then there would be gaps on the real line.

Last thing I didn’t mention that is interesting in this chapter is that cognitively we have a barrier to overcome with dimensionless points. There was a series of experiments in which children were told to shrink different shapes, triangles, circles, line segments, random polygons, etc to a point. It was carefully explained that this point should be infinitely small so as to lose all dimensions. Just a speck. But when they were told to visualize it growing in size, it went back to its original shape. This means that the way in which we visualize the construction of a point does affect our interpretation (cognitively) of what the “point” is.

Yes, people seem to means lots of things by “point.” An infinitely small speck can still have a shape (if there are infinitesimals). I always think of a point as round, even though that’s absurd (I guess they are always drawn as round). But presumably a dimensionless point is shapeless. Still, children probably have various ideas of what a dimension is. And if IT is growing again then it kind of makes sense for IT to reacquire its previous form (like rehydrating some powdered stuff). So I would say that points, as dimensionless parts of geometrical (paradigmatically quasi-spatial) entities, have a unique (since minimal) nature.

By “fill in the line with real numbers” do you mean that two arbitrary points are assigned the labels “0” and “1” and then the line is filled with points with real number labels? Then the rationals would not fill the line; and if the reals filled it then the hyperreals would be purely formal entities. If there really were hyperreals in the geometrical line (although there is Moore’s “sliding” problem (Synthese 113, December 2002)) then the points with real number labels would not have filled it. If there are physical continua (maybe time) then there would be a fact of the matter about what sort of labels could fully label the geometrical points, no?

I think the argument about filling in the line with rationals is talking about the fact that you have to consider the space you are living in. In the space of rationals there are no gaps between rational numbers. One thing I was thinking about is that the metaphor should work if you want to use it. Maybe I’m still mixing metaphors and it is becoming confusing, but the filling in metaphor doesn’t seem to work properly for conceptualizing what you actually want to have. The metaphor about placing points on the line just doesn’t even make sense to me right now if you consider the real numbers from 0 to 1. How can you drop the points on the line? It’s impossible since they are uncountable.

Conceptualizing “point” seems to me like conceptualizing a four dimensional space. Sure shrink something down infinitely small, but that is an iterative process that never ends. It isn’t a visualization of any sort like the kids were asked to do. By point they were thinking of something that still had dimension which I think should retain the structure if you blow it up or shrink it down a finite amount. The conceptualization of infinity seems to be a large sticking point for any student. And dimensionless and four dimensional space are both things that we do not encounter in real life, so they also are hard to conceptualize (not even going into visualizing).